Super-resolution optical fluctuation imaging
In comparison to other super-resolution microscopy techniques such as STORM or PALM that rely on single-molecule localization and hence only allow one active molecule per diffraction-limited area (DLA) and timepoint,[1][2] SOFI does not necessitate a controlled photoswitching and/ or photoactivation as well as long imaging times.
In mathematical terms SOFI-imaging relies on the calculation of cumulants, for what two distinguishable ways exist.
For one thing an image can be calculated via auto-cumulants[3] that by definition only rely on the information of each pixel itself, and for another thing an improved method utilizes the information of different pixels via the calculation of cross-cumulants.
[5] Both methods can increase the final image resolution significantly although the cumulant calculation has its limitations.
[3] Likewise to other super-resolution methods SOFI is based on recording an image time series on a CCD- or CMOS camera.
In contrary to other methods the recorded time series can be substantially shorter, since a precise localization of emitters is not required and therefore a larger quantity of activated fluorophores per diffraction-limited area is allowed.
The finally assigned pixel value intensities are a measure of the brightness and correlation of the fluorescence signal.
The cumulant calculation now filters the signal and leaves only highly correlated fluctuations.
As it is implied in the figure on the left the fluorescence source distribution: is convolved with the system's point spread function (PSF) U(r).
The molecular brightness is just the average fluorescence count-rate divided by the number of molecules within a specific region.
For simplification it has to be assumed that the sample is in a stationary equilibrium and therefore the fluorescence signal can be expressed as a zero-mean fluctuation: where
Using only the simple correlation function for a reassignment of pixel values, would ascribe to the independency of fluctuations of the emitters in time in a way that no cross-correlation terms would contribute to the new pixel value.
For computational reasons it is convenient to set all time-lags in higher-order cumulants to zero so that a general expression for the n-th order auto-cumulant can be found:[3]
is a specific correlation based weighting function influenced by the order of the cumulant and mainly depending on the fluctuation properties of the emitters.
Emitters with a higher molecular brightness will show a strong increase in terms of the pixel cumulant value assigned at higher-orders as well as this performance can be expected from a diverse appearance of fluctuations of different emitters.
[3][5] The calculation of auto-cumulants can be realized in a very attractive way in a mathematical sense.
This way of computation is straightforward in comparison with calculating cumulants with standard formulas.
In a more advanced approach cross-cumulants are calculated by taking the information of several pixels into account.
In addition the cross-cumulant approach can be used to estimate the PSF of the optical system by making use of the intensity differences of the virtual pixels that is due to the "loss" in cross-correlation as aforementioned.
At last the PSF can be used to create a resolution dependency of n for the nth-order cumulant by re-weighting the "optical transfer function" (OTF).
The cross-cumulant approach facilitates the generation of virtual pixels depending on the order of the cumulant as previously mentioned.
Part B of the second image depicts this general dependency of the virtual pixels on the cross-correlation.
